Differentiable maps with isolated critical points are not necessarily open in infinite dimensional spaces

TL;DR

This paper demonstrates that in infinite dimensional Hilbert spaces, differentiable maps with isolated critical points do not necessarily exhibit openness, providing counterexamples to a previously questioned assumption.

Contribution

The authors construct counterexamples showing that differentiable maps with isolated critical points are not always open in infinite dimensional spaces.

Findings

Counterexamples in Hilbert spaces show non-openness.

Differentiable maps with isolated critical points can fail to be open.

Answers a long-standing question negatively.

Abstract

Jean Saint Raymond asked whether continuously differentiable maps with isolated critical points are necessarily open in infinite dimensional (Hilbert) spaces. We answer this question negatively by constructing counterexamples in various settings.